-functions and distributions in this framework.

Kernel Calculus and Extension of Contact Transformations to D-Modules

Andrea D’Agnolo Pierre Schapira

1 Introduction

There is an important literature dealing with integral transformations. In our papers [6], [7] we proposed a general framework to the study of such transforms in the language of sheaves and D-modules. In particular, we showed that there are two natural adjunction formulas which split many dif- ficulties into two totally different kind of problems: one of analytical nature, the calculation of the transform of a D-module, the other one topological, the calculation of the transform of a constructible sheaf. Similar adjunction formulas for temperate and formal cohomology are obtained by M. Kashi- wara and P. S. in [15], and allow one to treat C

-functions and distributions in this framework.

Here, we shall first recall the four above mentioned adjunction formu- las, and then concentrate our study on the D-module theoretical transform.

Given two complex manifolds X and Y and a D-module kernel which defines a quantized contact transformation on an open subset of ˙ T

(X × Y ), our main result (Theorem 3.6 below) gives a geometrical condition to extend it as an isomorphism of locally free D-modules of rank one. This improves our previous result of [7].

These results apply to classical problems of integral geometry, in the line of Leray [16], Martineau [18], Gelfand-Gindikin-Graev [9], Helgason [10] or Penrose [8]. In particular, they allowed us to treat projective duality and the twistor correspondence. (Refer to [6], [7], [5]. See also [17] for other flag correspondences.)

Dedicated to Professor Hikosaburo Komatsu AMS classification: 30E20, 32L25, 58G37

Appeared in: “New trends in microlocal analysis” (Tokyo, 1995) Springer-Verlag,

1997, 179–190.

2 Review on the calculus of kernels

In this section we will develop the formalism of kernels in the framework of sheaves and D-modules. The results below concerning kernels for sheaves and D-modules are well-known from the specialists: let us mention in particular M. Kashiwara and also J.-P. Schneiders, with whom we had many discussions on this subject.

2.1 A review on sheaves, D-modules and temperate cohomology

References are made to [14] for the theory of sheaves, and to [19] and [11] for the theory of D-modules (see [22] for a detailed exposition).

Let X be a real analytic manifold, and denote by a

the map from X to the set consisting of a single element. We denote by D

(C

) the derived category of the category of bounded complexes of sheaves of C-vector spaces on a topological space X. If A ⊂ X is a locally closed subset, we denote by C

the sheaf on X which is the constant sheaf on A with stalk C, and zero on X \ A. We consider the “six operations” of sheaf theory RHom(·, ·),

· ⊗ ·, Rf

, Rf

, f

, f

, and we denote by × the exterior tensor product.

Recall that RHom(·, ·) = Ra

RHom(·, ·). For F ∈ D

(C

) we set D

F = RHom(F, C

), DF = RHom(F, ω

), where ω

' or

[dim

X ] denotes the dualizing complex, and or

the orientation sheaf.

We denote by SS(F ) the micro-support of F , a closed conic involutive subset of T

X. We denote by D

(C

) the full triangulated subcategory of D

(C

) of objects with R-constructible cohomology. If X is a complex manifold, one defines similarly the category D

(C

) of C-constructible objects.

Let X be a complex manifold of dimension d

. We denote by O

the structural sheaf, by Ω

the sheaf of holomorphic forms of maximal degree, and by D

the sheaf of rings of linear differential operators. We denote by Mod(D

) the category of left D

-modules, and by Mod

(D

) the full sub- category of Mod(D

) consisting of good D

-modules. This is the smallest thick subcategory of Mod(D

) containing the coherent modules which can be endowed with good filtrations on a neighborhood of any compact subset of X. Note that in the algebraic case, coherent D-modules are good. We denote by D

(D

) the derived category of the category of bounded com- plexes of left D

-modules, and by D

(D

) its full triangulated subcat- egory whose objects have cohomology groups belonging to Mod

(D

).

We consider the operations in the derived category of (left or right) D- modules: RHom

(·, ·), · ⊗

X

·, f

, f

, f

. In particular, if M ∈ D

(D

),

N ∈ D

(D

), and f : Y −→ X:

f

M = D

⊗

f

M, f

!

N = Rf

(D

⊗

Y

N ),

where D

and D

are the transfer bimodules associated to f . We denote by × the exterior tensor product, and we also use the notation:

D

M = RHom

(M, K

),

where K

denotes the dualizing complex for left D

-modules, defined by K

= D

⊗

Ω

[d

]. If F is a holomorphic vector bundle on X, we set:

F

= Hom

(F , O

), DF = D

⊗

F.

Let us briefly recall some constructions of [12] and [15].

First, assume X is a real analytic manifold. Denote by Db

the sheaf of Schwartz’s distributions on X, and by C

the sheaf of functions of class C

. There exist unique contravariant functors, exact for the natural t-structures:

T hom(·, Db

) : D

(C

)

−→ D

(D

),

· ⊗ C

: D

(C

) −→ D

(D

), such that if Z is a closed subanalytic subset of X, then

T hom(C

, Db

) = Γ

Db

, C

⊗ C

= I

,

where I

denotes the ideal of C

of functions vanishing to infinite order on Z.

Now, assume that X is a complex manifold. Denote by X the associated anti-holomorphic manifold, by X

the underlying real analytic manifold, and identify X

to the diagonal of X × X. For F ∈ D

(C

) one sets:

T hom(F, O

) = RHom

X

(O

, T hom(F, Db

)), F ⊗ O

= RHom

X

(O

, F ⊗ C

).

In other words, one defines T hom(F, O

) and F ⊗ O

as the Dolbeault complexes with coefficients in T hom(F, Db

), and F ⊗ C

respectively.

If F ∈ D

(C

), then T hom(F, O

) has regular holonomic cohomology groups (this is the way Kashiwara proves the Riemann-Hilbert equivalence of categories). In such a case, one has:

T hom(D

F, O

) ' D T hom(F, O

).

If Z is a closed complex submanifold of codimension d of X, we shall con- sider the holonomic left D

-module B

= T hom(C

[−d], O

) of [19]. Re- call that B

' H

(O

) (algebraic cohomology) is a subsheaf of B

= H

(O

).

2.2 Kernels for sheaves

Here, all manifolds and morphisms of manifolds will be complex analytic.

Let X and Y be complex manifolds of dimension d

and d

respectively.

Denote by r : X × Y −→ Y × X the map r(x, y) = (y, x), and by q

, q

the first and second projection from X × Y to the corresponding factor. If Z is another manifold, for i, j = 1, 2, 3 we denote by q

the projections from X × Y × Z to the corresponding factor (e.g., q

: X × Y × Z −→ Y × Z).

Definition 2.1. For K ∈ D

(C

) and L ∈ D

(C

), we set:

K ◦ L = Rq

(q

K ⊗ q

L ),

t

K = r

D

K.

Note that the operation ◦ is associative.

For K ∈ D

(C

), L ∈ D

(C

), consider the hypotheses:

(supp(K) × Z) ∩ (X × supp(L)) is proper over X × Z, (2.1) (SS(K) × T

Z) ∩ (T

X × SS(L)) ⊂ T

(X × Y × Z). (2.2) Proposition 2.2. Let K ∈ D

(C

), L ∈ D

(C

), and H ∈ D

(C

).

Assume (2.1), (2.2). Then:

RHom(H, K ◦ L) ' RHom((

K) ◦ H, L)[−2d

].

Proof. One has the chain of isomorphisms:

RHom(H, K ◦ L) = RHom(H, Rq

(q

K ⊗ q

L )) ' RHom(H, Rq

(q

K ⊗ q

L )) ' RHom(q

H, q

K ⊗ q

L)

' RHom(q

H, RHom(q

D

K, q

L)) ' RHom(q

H ⊗ q

D

K, q

L)

' RHom(q

H ⊗ q

D

K, q

L)[−2d

] ' RHom(Rq

(q

H ⊗ q

D

K[2d

]), L)

= RHom((

K ) ◦ H, L)[−2d

].

Here, we used hypothesis (2.1) in the first isomorphism, and hypothesis (2.2)

in the third.

Corollary 2.3. Let K ∈ D

(C

), and assume supp(K) is proper over X,

SS (K) ∩ (T

X × T

Y ) ⊂ T

(X × Y ).

Then there are natural morphisms:

C

X

−→ K ◦

K [2d

], K ◦

K [2d

] −→ C

.

Proof. Applying Proposition 2.2 for Z = X, H = C

, L =

K we obtain the first morphism. Choosing instead Z = Y , L = C

, H = K, we get a morphism

K ◦ K[2d

] −→ C

, from which the second morphism in the statement is easily deduced.

Assuming (2.1), (2.2), and assuming that K or L is R-constructible, one proves similarly that

D

(K ◦ L) ' D

K ◦ D

L [2d

].

2.3 Kernels for D-modules

Definition 2.4. For K ∈ D

(D

) and L ∈ D

(D

), we set:

K ◦ L = q

(q

K ⊗

X ×Y ×Z

q

L) Note that

K ◦ L ' q

δ

(K × L),

where δ denotes the diagonal embedding X × Y × Z −→ X × Y × Y × Z.

Proposition 2.5.

Let K ∈ D

(D

) and N ∈ D

(D

). Then, there is a natural isomorphism in D

(D

):

K ◦ N ' Rq

(K

⊗

2 DY

q

N ), where K

= K ⊗

2 OY

q

Ω

is endowed with its natural (q

D

, q

D

)- bimodule structure.

1

As pointed out to us by Andrei Baran, Proposition B.5 of [7] holds only in the algebraic

case. In the analytic case, it should be replaced by Proposition 2.5 above.

Proof. By definition,

K ◦ N ' Rq

((D

× Ω

) ⊗

X ×Y

(K ⊗

X ×Y

(O

× N ))) Moreover, one has the following chain of isomorphisms:

(D

× Ω

) ⊗

X ×Y

(K ⊗

X ×Y

(O

× N )) ' q

Ω

⊗

2 DY

(K ⊗

2 OY

q

N ) ' (q

Ω

⊗

2 OY

K) ⊗

2 DY

q

N ' K

⊗

2 DY

q

N .

Proposition 2.6. Let K ∈ D

(C

), L ∈ D

(C

), and assume (2.1). Then there is a natural isomorphism:

T hom(K, O

) ◦ T hom(L, O

) −→ T hom(K ◦ L, O

)[−d

].

Proof. Consider the chain of isomorphisms:

T hom(K, O

) ◦ T hom(L, O

)

= q

(q

T hom(K, O

) ⊗

X ×Y ×Z

q

T hom(L, O

)) ' q

(T hom(q

K, O

) ⊗

X ×Y ×Z

T hom(q

L, O

)) ' q

T hom(q

K ⊗ q

L, O

)

' T hom(Rq

(q

K ⊗ q

L ), O

)[−d

]

= T hom(K ◦ L, O

)[−d

].

For the proof of the above isomorphisms, see [12], [1], [15], and [3]. Note that we used hypothesis (2.1) in the third isomorphism.

Proposition 2.7. For K ∈ D

(D

) and L ∈ D

(D

), consider the analogous hypotheses to (2.1) (2.2):

(supp(K) × Z) ∩ (X × supp(L)) is proper over X × Z, (2.3) (char(K) × T

Z) ∩ (T

X × char(L)) ⊂ T

(X × Y × Z). (2.4) Then there is a natural isomorphism:

D(K ◦ L) ' DK ◦ DL. (2.5)

Proof. Under the above assumptions, D commutes to the operations appear- ing in the definition of ◦.

Proposition 2.8. Let K be a regular holonomic D

-module, and let K be the associated perverse sheaf K = RHom

(K, O

). Set

t

K = r

DK ' T hom(

K, O

).

Assume (2.1) and (2.2) (or equivalently (2.3) and (2.4)) with Z = X and L =

K (or equivalently L =

K). Then there are natural morphisms:

K ◦

K[d

− d

] −→ B

,

B

−→ K ◦

K[d

− d

].

Proof. Applying Corollary 2.3, we get the morphism:

T hom(C

[−d

], O

) ←− T hom(K ◦

K[d

], O

)[d

− d

].

The first morphism follows by applying Proposition 2.6. The second mor- phism is similarly obtained.

Remark 2.9. Consider a correspondence:

X ←− S

−→ Y,

(2.6)

and denote by h : S −→ X ×Y the morphism h = (f, g). It is then immediate to check that for F ∈ D

(C

), K ∈ D

(C

), one has:

F ◦ (Rh

K ) ' Rg

(K ⊗ f

F ).

Moreover, assuming h is proper it is easy to check that for M ∈ D

(D

), K ∈ D

(D

):

M ◦ (h

K) ' g

!

(K ⊗

S

f

M).

Recall that, in the particular case when h is a closed embedding, one has h

O

' B

.

2.4 Adjunction formulas

Formulas (2.10), (2.11) below appeared in a slightly more particular situation in [6]. Formulas (2.12), (2.13) are due to [15].

For K a regular holonomic D

-module, set K = RHom

(K, O

).

Consider the hypotheses:

(supp(M) × Y ) ∩ supp(K) is proper over Y, (2.7)

(char(M) × T

Y ) ∩ char(K) ⊂ T

(X × Y ), (2.8)

(X × supp(G)) ∩ supp(K) is proper over X. (2.9)

Theorem 2.10.

(i) Assuming hypotheses (2.7) and (2.8) above, we have isomorphisms:

RΓ

(X; RHom

(M, (K ◦ G) ⊗ O

))[d

] (2.10) ' RΓ

(Y ; RHom

(M ◦ K, G ⊗ O

)),

RΓ(X; RHom

(M, RHom(G ◦

K, O

)))[d

] (2.11) ' RΓ(Y ; RHom

(M ◦ K, RHom(G, O

)))[2d

].

If moreover (2.9) is satisfied, the formulas above hold interchanging Γ and Γ

.

(ii) Assuming hypotheses (2.7) and (2.9) above, we have an isomorphism:

RΓ(X; RHom

(M, (K ◦ G) ⊗ O

))[d

] (2.12) ' RΓ(Y ; RHom

(M ◦ K, G ⊗ O

)).

If moreover (2.8) holds, then we have an isomorphism:

RΓ

(X; RHom

(M, T hom(G ◦

K, O

)))[d

] (2.13) ' RΓ

(Y ; RHom

(M ◦ K, T hom(G, O

)))[2d

].

Under the same hypotheses, formulas (2.12) and (2.13) hold interchanging Γ and Γ

.

3 Generalized QCTs

3.1 Kernels for E-modules

We recall here some definitions from the theory of E-modules. We refer to [19]

and to [20] for an exposition.

We denote by π

: T

X −→ X the cotangent bundle to X, by ˙π

: T ˙

X −→ X the cotangent bundle with the zero-section removed, and by T

X the conormal bundle to a submanifold M of X. For a subset V of T

X , we set ˙ V = V ∩ ˙ T

X . We denote by p

and p

the first and second projection from T

(X × Y ) ' T

X × T

Y to the corresponding factor, and by p

the composite of p

with the antipodal map of T

Y .

Let E

denote the sheaf of microdifferential operators of finite order on T

X. We denote by D

(E

) the derived category of the category of bounded complexes of left E

-modules.

2

In formulas (C.4) and (C.7) of [7], K ◦ G should be replaced by G ◦

K as in formulas

(2.11) and (2.13) above.

To f : Y −→ X one associates the natural maps:

T

Y ←−

tf⁰

Y ×

T

X −→

fπ

T

X.

We will denote by f

and f

µ

the inverse and direct images in the sense of E-modules. Hence, for M ∈ D

(E

) and N ∈ D

(E

):

f

M = R

f

(E

⊗

π EX

f

M), f

µ

N = Rf

(E

⊗

t

f

N ), where E

and E

are the transfer bimodules.

If M ∈ D

(D

), we set:

EM = E

⊗

X DX

π

M,

an object of D

(E

), and if F is a holomorphic vector bundle on X, we set:

EF = E(DF).

If Z is a closed complex submanifold of codimension d of X, we shall consider the holonomic left E

-module C

= EB

.

Definition 3.1. Let K ∈ D

(D

) and L ∈ D

(D

). Denoting by δ : X × Y × Z −→ X × Y × Y × Z the diagonal embedding, we set:

EK ◦

EL = q

δ

(EK × EL).

Proposition 3.2. (i) Let M ∈ D

(D

), and assume that f : Y −→ X is non-characteristic for M. Then

Ef

M ' f

EM.

(ii) Let N ∈ D

(D

). Assume that f is proper on supp N . Then Ef

N ' f

µ

EN . (iii) Let M ∈ D

(D

), N ∈ D

(D

). Then

E(M × N ) ' (EM) × (EN ).

Proof. Assertion (iii) is obvious, and (ii) is proved in [21]. Assertion (i)

follows from the division theorem of [19], using the techniques of that paper.

From Proposition 3.2 it immediately follows:

Corollary 3.3. Assuming (2.3), (2.4), there is a natural isomorphism:

E(K ◦ L) ' EK ◦

EL.

Theorem 3.4. Let K ∈ D

(D

), N ∈ D

(D

). Assume:

supp(K) is proper over X,

char(K) ∩ (T

X × T

Y ) ⊂ T

(X × Y ).

Then there is a natural isomorphism:

EK ◦

EN −→ Rp

(EK

⊗

−1EY

p

EN ).

Proof. By Corollary 3.3, the left hand side is isomorphic to E(K ◦ N ). Hence, by Proposition 2.5, it is enough to prove the natural isomorphism:

E

Rq

(K ⊗

Y

N ) −→ Rp

(E

K ⊗

Y

N ).

Let E

denote the subsheaf of E

of sections which commute with O

. The second hypothesis, and the division theorem of [19] gives:

E

K ' E

K

Let K

be a coherent O

-module which generates K, and N

be a coherent O

-module which generates N . Let L

= K

⊗

N

. It is enough to check:

E

(0) ⊗

Rp

L

−→ Rp

(E

(0) ⊗

L

). (3.1) E

(0) is an O

-module of type DFN and E

(0) ' E

(0) b ⊗O

(see [21, I §7]). Then the proof goes as that of Theorem 7.3 of [15], using Proposition 3.13 of [21] (or Theorem 8.1 of [15]).

3.2 Extending QCTs to D-modules

In this section, we assume X and Y have the same dimension n. Let L be a regular holonomic D

-module, and set:

L = RHom

(L, O

), Λ = char(L) ⊂ T

(X × Y ).

Let F and G be holomorphic line bundles on X and Y respectively, and set:

L

(F , G) = q

(F ⊗

Ω

) ⊗

1 OX

L ⊗

2 OY

q

G.

Lemma 3.5. Assuming q

is proper on supp L, there is a natural isomor- phism:

α : Γ(X × Y ; L

(F , G

)) ' Hom

(DG, DF ◦ L). (3.2) Proof. It is enough to apply H

(·) to the following chain of isomorphisms:

Ra

L

(F , G

) ' ' Ra

Rq

RHom

2 DY

(q

DG, L

⊗

1 OX

q

F) ' Ra

RHom

(DG, Rq

(L

⊗

1 OX

q

F)) ' Ra

RHom

(DG, DF ◦ L).

Here, in the first isomorphism we used the fact that DG is D

-coherent, and in the last one we used the fact that q

is proper on supp L.

Assuming q

is proper on supp L, by Lemma 3.5 we associate to s ∈ Γ(X × Y ; L

(F , G

)) the D

-linear morphism:

α (s) : DG −→ DF ◦ L.

Let U

and U

be two open conic subsets of ˙ T

X and ˙ T

Y respectively.

Set

Λ

= Λ ∩ (U

× U

), Σ = Λ \ Λ

, W = T

Y \ U

. We assume:

(a) ˙ W is a C-analytic closed conic subset of ˙ T

Y of codimension ≥ 2, (b) Λ

is a smooth Lagrangian manifold, and p

: Λ

−→ U

, p

: Λ

−→

U

are isomorphisms (in other words, Λ

defines a contact transforma- tion),

(c) p

W ˙ ∩ ˙Λ = ˙Σ, and this analytic set has dimension < n, (d) L has no submodules isomorphic to O

,

(e) there exists s ∈ Γ(X × Y ; L

(F , G

)), which is non-degenerate on Λ

.

We refer to [19] for the notion of non-degenerate section of a holonomic module.

Theorem 3.6. Assume hypotheses (a)–(e) above. Then:

H

(α(s)) : DG −→ H

(DF ◦ L)

is an isomorphism.

Proof. Applying Corollary 3.3, it is enough to prove the isomorphism EG −→

H

(EF ◦

EL) all over T

Y . Consider the morphism of distinguished trian- gles:

RΓ

(EG) −→ Rπ

(EG) −→ RΓ

(EG) −→

↓

↓

↓

RΓ

(EF ◦

EL) −→ Rπ

(EF ◦

EL) −→ RΓ

(EF ◦

EL) −→

(3.3) By (b) and (c), RΓ

(EF ◦

EL) ' RΓ

(EF |

◦

EL|

). Since s is non-degenerate on Λ

, α

(s) is an isomorphism by [19]. Applying H

(·), we get the commutative diagram in which the horizontal lines are exact:

H

EG → DG → H

EG −→ H

EG

↓ ↓

↓

↓

H

(EF ◦

EL) → H

(DF ◦ L) → H

(EF ◦

EL) −→ H

(EF ◦

EL).

We shall prove (i) H

EG = 0,

(ii) H

(EF ◦

EL) = 0, (iii) H

EG = 0.

This will imply the result, for then β will be the zero morphism.

Since the problem is local on T

Y , in (i) and (iii) we may assume G = O

. Then (i) and (iii) follow from [13, Theorem 1.2.2]. To prove (ii), consider the isomorphisms:

RΓ

(EF ◦

EL) ' RΓ

Rp

(EL

⊗

1 OX

p

F) ' Rp

RΓ

(EL

⊗

1 OX

p

F).

We thus reduce to a local problem on Λ. Applying H

(·), it is enough to prove that

Γ

EL = 0.

Since Σ contains the zero-section, using the exact sequence:

0 −→ Γ

EL −→ Γ

EL −→ Γ

EL,

and recalling hypothesis (d), it remains to prove that Γ

EL = 0. Since ˙Σ is an analytic subset of dimension smaller than that of X × Y , it follows by the involutivity theorem that L has no germs of sections supported by ˙Σ.

Remark 3.7. (i) In [7] we obtained a similar result in the case where Λ

= ˙Λ, a situation which applies to projective duality.

(ii) Other applications of Theorem 3.6 will be found in [17].

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[3] J-E. Bj¨ork, Analytic D-modules and Applications. Kluwer Academic Pub- lisher, Dordrecht-Boston-London (1993)

[4] J. L. Brylinski, Transformations canoniques, dualit´e projective, th´eorie de Lefschetz, transformations de Fourier et sommes trigonom´ etriques, Ast´erisque 140-141 (1986), 3–134.

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